Nonlinear wave equations perturbed by viscous terms. Transl. from the Russian by M. A. Shishkova
type: Item
instance of: scholarly article
title: Nonlinear wave equations perturbed by viscous terms. Transl. from the Russian by M. A. Shishkova
zbMATH Open document ID: 0951.35002
author: V. P. Maslov
author: P. P. Mosolov
published in: De Gruyter Expositions in Mathematics
publication date: 2000-08-20T00:00:00Z
review text: This monograph is devoted to a wide range of interesting mathematical problems arising in nonlinear continuum mechanics, such as nonstationary flow of viscous gas and dynamic deformation of media with different rigidity in tension and compression. The authors focus their attention on a classical system of equations describing the motion of a viscous one-dimensional compressible fluid \[ \rho _t+(\rho u)_x =0,\quad u_t+uu_x=\rho^{-1}(-p_x +\mu u_{xx}),\tag{1} \] where \(p=-\varphi (1/\rho)\) ( \(\varphi '>0\)) is the pressure, \(v=1/\rho \) is the specific volume, and \(\mu \) is a small viscosity parameter. By Lagrangian coordinates the system (1) is transformed in \[ u_{tt}-(\varphi (u_x))_x-\mu (\psi (u_x))_{xt}=0,\tag{2} \] where \(\psi (\lambda)=\log (\lambda)\). This is a hyperbolic equation with artificial viscosity. Necessary and sufficient conditions for the existence of smooth (for vanishing viscosity) solutions of the problem under consideration are obtained. The book consists of 12 chapters. In Chapter 1 the authors obtain an interesting existence theorem for solutions of the Cauchy problem for an infinite system of ordinary differential equations describing a mechanical system of infinitely many particles on a straight line under viscous and elastic forces and viscous friction. An important role is played by a small parameter \(h\) determining the distance between particles. Some interesting a priori estimates of the solutions are obtained in Chapter 2. The known Voigt model \[ u_{tt}-\varphi (u_x)_x - \mu \psi (u_x)_{xt}=0 \] (\(\varphi '(a)>0\), \(\psi '(a)>0\)) is obtained after passing to the limit as \(h\to 0\). Some \textit a priori \textrm estimates are obtained under the assumption that the functions \(\varphi \) and \(\psi \), as well as their first- and second-order derivatives, are bounded. The existence and uniqueness of a generalized solution for more general hyperbolic equations \[ u_{tt}-\varphi (u_x)_x - \mu \psi (u_x)_{xt}+p(u)+q(u)=f(x,…
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 76N15
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 1493056
zbMATH Keywords: one-dimensional compressible gas
zbMATH Keywords: Navier-Stokes system
zbMATH Keywords: asymptotic solution
zbMATH Keywords: viscous compressible fluid
zbMATH Keywords: Lagrangian coordinates
zbMATH Keywords: Voigt model
zbMATH Keywords: Maxwell model
reviewed by: Dimitar A. Kolev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1493056